for primes p ≡ 1 (mod 3) up to 500, and found that Sp/(2√p) lay in the intervals [−1,− 12 ], (− 12 , 12 ), [ 1 2 , 1] with frequencies approximately in the ratio 1 : 2 : 3. He conjectured, somewhat hesitantly, that this might be true asymptotically. Kummer’s conjecture was disproved by Heath-Brown and Patterson [4]. In order to state their result we must introduce a little notation. Let ω = exp...

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...

We consider Gauss sums of the form Gn(a) = ∑ x∈Fpm χ(x) with a nontrivial additive character χ 6= χ0 of a finite field Fpm of pm elements of characteristic p. The classical bound |Gn(a)| ≤ (n−1)pm/2 becomes trivial for n ≥ pm/2 + 1. We show that, combining some recent bounds of HeathBrown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on |Gn(a)| which is...

We consider Gauss sums associated to functions T → R/Z which satisfy some sort of “quadratic” property and investigate their elementary properties. These properties and a Gauss sum formula from the nineteenth century due to Dirichlet enable us to give elementary proofs of many standard results. We derive the Milgram Gauss sum formula computing the signature mod 8 of a non– singular bilinear for...

We describe a family of quantum states the Schr\"odinger cat type as superpositions harmonic oscillator coherent with coefficients defined by quadratic Gauss sums. These emerge eigenfunctions lowering operators obtained after canonical transformations Heisenberg-Weyl algebra associated ordinary and fractional Fourier transformation. The first member this is given well known Yurke-Stoler state.

Let p be an odd prime, and let m divide p−1. Let ζ = e and let ω = e. The field extension Q(ω) ⊂ Q(ω, ζ) is Galois with cyclic Galois group isomorphic to (Z/pZ)×. The unique field between Q(ω) and Q(ω, ζ) having degree m over Q(ω) takes the form Q(ω, τ) where τ is a Gauss sum, to be described below. Furthermore, under some conditions we can compute τ as an element α of Q(ω), thus expressing the...

We define certain natural finite sums of n’th roots of unity, called GP (n), that are associated to each convex integer polytope P , and which generalize the classical 1-dimensional Gauss sum G(n) defined over Z/nZ, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group W, generated by the reflections with respect to the coordinate hyperplanes, as well as ...